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Simple numbers question... Or maybe not so simple.

The decimal numeration system is efficient, because it uses every combination of symbols. It is more efficient that Roman Numerals or other obsolete systems.

For example, if we have the symbols 0 to 9 ("0", "1", "2", "3", "4", "5", "6", "7", "8", "9") and n places, there are 10^n combinations, and 10^n different, consecutive, integer numbers, which can be represented.

Example:

"0" to "9" are 10 symbols, and for n=1, there are 10 numbers. For n=2 there are 100 numbers.

But the system efficiency is ruined when we have decimals, or negative numbers. With these we have 2 extra symbols, the "." and the "-".

The "-" can only be placed once, and only on one position. Meanwhile the "." can be on any place, but can be present only once.

That's wasteful. We have 12 symbols, but we can´t represent 12^n different numbers.

The problem is: Propose a numeration system able to represent 12^n different numbers, including any number traditionally represented with the symbols "-", ".", "0", "1", "2", "3", "4", "5", "6", "7", "8", "9" on n places.

For example, if we have the symbols 0 to 9 ("0", "1", "2", "3", "4", "5", "6", "7", "8", "9") and n places, there are 10^n combinations, and 10^n different, consecutive, integer numbers, which can be represented.

Example:

"0" to "9" are 10 symbols, and for n=1, there are 10 numbers. For n=2 there are 100 numbers.

But the system efficiency is ruined when we have decimals, or negative numbers. With these we have 2 extra symbols, the "." and the "-".

The "-" can only be placed once, and only on one position. Meanwhile the "." can be on any place, but can be present only once.

That's wasteful. We have 12 symbols, but we can´t represent 12^n different numbers.

The problem is: Propose a numeration system able to represent 12^n different numbers, including any number traditionally represented with the symbols "-", ".", "0", "1", "2", "3", "4", "5", "6", "7", "8", "9" on n places.

We can represent aleph-null different numbers. But so could the romans and the greeks and the hebrews.

ruveyn

But the system efficiency is ruined when we have decimals, or negative numbers. With these we have 2 extra symbols, the "." and the "-".

The "-" can only be placed once, and only on one position. Meanwhile the "." can be on any place, but can be present only once.

That's wasteful. We have 12 symbols, but we can´t represent 12^n different numbers.

.

You have confused number (the mathematical object) with numeral (they way numbers are written out).

You evaluation of negation and showing the fractional parts of real or rational numbers is just plain wrong.

ruveyn

The "-" can only be placed once, and only on one position. Meanwhile the "." can be on any place, but can be present only once.

That's wasteful.

That isn't really wasteful. To represent numbers that are negative or fractional, we need something like the negative sign and the decimal sign. In any representation, something must take their place or we must give up representing that type of number. We even have reasonable rules for omitting them in most common cases, so they only take up space when they give us information.

If you just want a more compact representation, look into non-base 10 number systems.

_________________

"A dead thing can go with the stream, but only a living thing can go against it." --G. K. Chesterton

The "-" can only be placed once, and only on one position. Meanwhile the "." can be on any place, but can be present only once.

That's wasteful.

That isn't really wasteful. To represent numbers that are negative or fractional, we need something like the negative sign and the decimal sign. In any representation, something must take their place or we must give up representing that type of number. We even have reasonable rules for omitting them in most common cases, so they only take up space when they give us information.

If you just want a more compact representation, look into non-base 10 number systems.

You bet. the minus sign (-) effectively doubles the range of the integers and the decimal point gets us from the integers to all the rational numbers. Infinite non-repeating decimals gets us to ALL the real numbers.

ruveyn

That's wasteful. We have 12 symbols, but we can´t represent 12^n different numbers.

You can get rid of the need to use the minus sign to represent negative numbers by having digits with negative values. IIRC, at some point during the Cold War, a computer was made in the USSR that worked with the balanced ternary system, whose digits represent -1, 0 and 1, and whose positions are weighted with powers of 3. This way, e.g., -19 is represented as 1̅101̅, and 33 as 111̅0. This strategy can be used with any odd number of symbols. With 11—corresponding to the 10 decimal digits plus something to replace the minus sign—you have 5̅, 4̅, 3̅, 2̅, 1̅, 0, 1, 2, 3, 4 and 5.

I don’t know if you can “efficiently” get rid of the radix point (that’s how you call the character separating the integer and fractional parts without referring to a specific base), because, in a positional system where digits are weighted with all integral powers of the base, you need a way to mark the place whose weight is 1 (i.e. whose exponent is 0). There may be ways to work around this, but they’re likely to involve more complexity than a humble dot or comma

EDIT – The digits with negative values should have overlines, but they don’t seem to show up

Well, they do show up to me if I copy the text from the forum and paste it elsewhere

_________________

The red lake has been forgotten. A dust devil stuns you long enough to shroud forever those last shards of wisdom. The breeze rocking this forlorn wasteland whispers in your ears, “Não resta mais que uma sombra”.

Last edited by Spiderpig on 05 Jun 2013, 2:35 pm, edited 1 time in total.

That's wasteful. We have 12 symbols, but we can´t represent 12^n different numbers.

You can get rid of the need to use the minus sign to represent negative numbers by having digits with negative values. IIRC, at some point during the Cold War, a computer was made in the USSR that worked with the balanced ternary system, whose digits represent -1, 0 and 1, and whose positions are weighted with powers of 3. This way, e.g., -19 is represented as 1̅101̅, and 33 as 111̅0. This strategy can be used with any odd number of symbols. With 11—corresponding to the 10 decimal digits plus something to replace the minus sign—you have 5̅, 4̅, 3̅, 2̅, 1̅, 0, 1, 2, 3, 4 and 5.

I don’t know if you can “efficiently” get rid of the radix point (that’s how you call the character separating the integer and fractional parts without referring to a specific base), because, in a positional system where digits are weighted with all integer powers of the base, you need a way to mark the place whose weight is 1 (i.e. whose exponent is 0). There may be ways to work around this, but they’re likely to involve more complexity than a humble dot or comma

EDIT – The digits with negative values should have overlines, but they don’t seem to show up

Well, they do show up to me if I copy the text from the forum and paste it elsewhere

Doing arithmetic with numeral sequences that have a mix of "positive digits" and "negative digits" would become painfully complicated. One minus sign to the left takes care of the matter a binary minus in the same position that could have been occupied by "+" does the trick. What you are suggesting would lead to unnecessary complication. If you really like overbars that much, then put an overbar over the entire number that is intended to be negative.

ruveyn

I don’t think it’s so complicated, but, at any rate, I was answering the OP’s question.

After thinking a bit more about this, the radix point can indeed be done away with easily: instead of assigning the weights the usual way, as powers of the base with exponents ranging from right to left, from -∞ to +∞, make the exponents increase in absolute value every other place from left to right, alternating signs, i.e. 0, 1, -1, 2, -2, 3, -3, etc.

The only real drawback I see to balanced numeral systems is that their bases have to be odd. A good one might be 15, as it has two prime factors, just like 10, and isn’t ridiculously large—you have to go to 105 to add a third divisor. Using colors to make it easier to tell the negative digits apart from the positive ones, you have 7̅, 6̅, 5̅, 4̅, 3̅, 2̅, 1̅, 0, 1, 2, 3, 4, 5, 6 and 7.

With base 15, the weights are 1, 15, 1/15, 225, 1/225, 3375, 1/3375 and so on. For example, 326.16 is represented as 4̅7216, -827.6666… (“6” recurring) as 3̅554̅, and 230.5 as 507170707… (“7” recurring at the places with fractionary weights) or 607̅17̅07̅07̅… (“7̅” recurring the same way).

As the last example shows, every rational number which can be expressed as an irreducible fraction with a denominator of the form 2 × 3^a × 5^b has two representations, but the standard decimal system has the same artifact with denominators of the form 2^a × 5^b, a.k.a. terminating decimals: 236.74 = 236.7399999… (“9” recurring).

PS – Oddly enough, the overlines seem to work just fine within the scope of a “[color]” tag.

_________________

The red lake has been forgotten. A dust devil stuns you long enough to shroud forever those last shards of wisdom. The breeze rocking this forlorn wasteland whispers in your ears, “Não resta mais que uma sombra”.

Last edited by Spiderpig on 05 Jun 2013, 4:09 pm, edited 1 time in total.

I guess there’s something wrong with my system then—sorry. I was afraid nobody would be able to read them correctly.

By the way, despite 15 being a base greater than 10, the balanced system makes it easier to memorize the multiplication tables

2 × 2 = 4

2 × 3 = 6

2 × 4 = 7̅1

2 × 5 = 5̅1

2 × 6 = 3̅1

2 × 7 = 1̅1

3 × 3 = 6̅1

3 × 4 = 3̅1

3 × 5 = 01

3 × 6 = 31

3 × 7 = 61

4 × 4 = 11

4 × 5 = 51

4 × 6 = 6̅2

4 × 7 = 2̅2

5 × 5 = 5̅2

5 × 6 = 02

5 × 7 = 52

6 × 6 = 62

6 × 7 = 3̅3

7 × 7 = 43

The tables for negative digits are the same as those of their opposites, replacing all the digits of the products with their opposites.

_________________

The red lake has been forgotten. A dust devil stuns you long enough to shroud forever those last shards of wisdom. The breeze rocking this forlorn wasteland whispers in your ears, “Não resta mais que uma sombra”.

That's wasteful. We have 12 symbols, but we can´t represent 12^n different numbers.

You can get rid of the need to use the minus sign to represent negative numbers by having digits with negative values. IIRC, at some point during the Cold War, a computer was made in the USSR that worked with the balanced ternary system, whose digits represent -1, 0 and 1, and whose positions are weighted with powers of 3. This way, e.g., -19 is represented as 1̅101̅, and 33 as 111̅0. This strategy can be used with any odd number of symbols. With 11—corresponding to the 10 decimal digits plus something to replace the minus sign—you have 5̅, 4̅, 3̅, 2̅, 1̅, 0, 1, 2, 3, 4 and 5.

Yes. That's an excellent answer to get rid of the minus sign.

For example, if base is -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,

which we can represent with the symbols B, C, D, A, 0, 1, 2, 3, 4, 5 or for clarity: 4, 3, 2, 1, 0, 1, 2, 3, 4, 5

Then, for example, the numbers -20 to 20 could get represented as:

-20 = 2 0

-19 = 2 1

-18 = 2 2

-17 = 2 3

-16 = 2 4

-15 = 2 5

-14 = 1 4

-13 = 1 3

-12 = 1 2

-11 = 1 1

-10 = 1 0

-9 = 1 1

-8 = 1 2

-7 = 1 3

-6 = 1 4

-5 = 1 5

-4 = 4

-3 = 3

-2 = 2

-1 = 1

0 = 0

1 = 1

2 = 2

3 = 3

4 = 4

5 = 5

6 = 1 4

7 = 1 3

8 = 1 2

9 = 1 1

10 = 1 0

11 = 1 1

12 = 1 2

13 = 1 3

14 = 1 4

15 = 1 5

16 = 2 4

17 = 2 3

18 = 2 2

19 = 2 1

20 = 2 0

For example,

21 would mean -20 + 1 =-19

14 would mean -10 - 4=-14

Last edited by Buxcador on 05 Jun 2013, 8:55 pm, edited 2 times in total.

ruveyn

No, it would be much simpler to use -4 to 5 than 0 to 9

For example, for multiplication is necessary to remember the tables from 0 to 9, but the negative base I used as example, only needs 0 to 5 multiplication tables, because the tables for negatives are similar to the positives.

Addition also would be simplified. Children would learn addition and multiplication much faster. Schools would spend much less money, and more people would like math, because it would require less effort.

Last edited by Buxcador on 05 Jun 2013, 8:02 pm, edited 1 time in total.

I suspect that is fair and convenient to assume that 7=07=000007. I other words, any number is supposed to have any quantity of zeros to the left.

After thinking a bit more about this, the radix point can indeed be done away with easily: instead of assigning the weights the usual way, as powers of the base with exponents ranging from right to left, from -∞ to +∞, make the exponents increase in absolute value every other place from left to right, alternating signs, i.e. 0, 1, -1, 2, -2, 3, -3, etc.

I think that it is just reordering the digits.

The common convention is to use ... -3, -2, -1, 0, 1, 2, 3,...

For example, this number 1234.56789 would be written as 43526170809, but that would not be more efficient, because it just sorts traditional representation, and adds lots of zeros.

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